A telephone wire stretched tight between two poles placed ten meters apart is a favorite resting spot for a flock of crows.

Suppose one morning two crows land on the wire, each at a random spot (the probability is uniformly distributed). With a bucket of paint and a brush you mark the stretch of wire between them. A certain length of wire will have been painted.

On average, what length of wire would you expect to have painted? Assume that each bird is a single point along the line, and so has no width.

Suppose instead that a dozen crows landed on the wire, each at an independent, random location, and you painted the stretch of wire between each bird and its nearest neighbor. On average, what total length of wire would you expect to have painted now?

And if a thousand crows landed?

A computer-generated solution could be found, but bonus points will be awarded for a formal proof!

let's say that the question says that the wire is stretched between 2
poles tight right that means that it's in a stright line and has no
bending well good for the birds .

Now
if i get the question correctly then it says that the birds have no width and hence they are a point as said in the question.

Now there is the thing

if i rember correctly then a stright line has infinite points and hence an infinite amount of birds can land on the line.

Possible situation I

infinite amount of birds land on the wire and hence cover each and every point on the line

in this case the ans will be 0

Possible Situation II

as said in the question 2 birds land on the wire haveing infinite points

in this case the ans is basec on randomness let's say
the point they land on is 5 80 :) not too big numbers considering
that the choice is from 1 - infinite :) not then the Avg. will be
42.5 if i remember the maths :) now consider that

2 birds landed on the wire again now again if we take 2 random points
say 20 106 then now the Avg. will be 63 sa we can see the ans will
differ from here and their and if we take an Avg. of the ans :) then we
get 52.75 even this number will varry from here to their.

Possible Situation III

this one i just came up with while writing lol
now consider if the 2 birds landed at the most extreme points in the line i.e at the start and the end

i.e
one at 1 and another at infinity then the Avg. will be 1 + infinity/2

Conclusion: Well i just want to say that the Avg. will differ from 1.5
to (infinity - 1) + infinity/2 that is what i can come up with in
mearly few mins lol :) well at least mail me the ans i realy would like
to know